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Totally multiplicative sequence with a(p) = 8*(p-3) for prime p.
1

%I #11 Oct 22 2023 00:50:07

%S 1,-8,0,64,16,0,32,-512,0,-128,64,0,80,-256,0,4096,112,0,128,1024,0,

%T -512,160,0,256,-640,0,2048,208,0,224,-32768,0,-896,512,0,272,-1024,0,

%U -8192,304,0,320,4096,0,-1280,352,0,1024,-2048,0,5120,400,0,1024,-16384

%N Totally multiplicative sequence with a(p) = 8*(p-3) for prime p.

%H G. C. Greubel, <a href="/A167318/b167318.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = (8*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)-3))^e(k).

%F a(3k) = 0 for k >= 1.

%F a(n) = A165829(n) * A166589(n) = 8^bigomega(n) * A166589(n) = 8^A001222(n) * A166589(n).

%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* _G. C. Greubel_, Jun 09 2016 *)

%t f[p_, e_] := (8*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 22 2023 *)

%Y Cf. A001222, A165829, A166589.

%K sign,easy,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009